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We present the results of theoretical researches of the developed hyperspherical function
for the appropriate functional matrix, generalized on the basis of two degrees of
freedom, and , and the radius . The precise analysis of the hyperspherical matrix for the field of natural numbers,
more specifically the degrees of freedom, leads to forming special translators that
connect functions of some hyperspherical and spherical entities, such as point, diameter,
circle, cycle, sphere, and solid sphere

1. Introduction

The hypersphere function is a hypothetical function connected to multidimensional
space. It belongs to the group of special functions, so its testing is performed on
the basis of known functions such as the -gamma, -psi, -beta, anderf-error function. The most significant value is in its generalization from discrete
to continuous. In addition, we can move from the scope of natural integers to the
set of real and noninteger values. Therefore, there exist conditions both for its
graphical interpretation and a more concise analysis. For the development of the hypersphere
function theory see Bishop and Whitlock [1], Collins [2], Conway and Sloane [3], Dodd and Coll [4], Hinton [5], Hocking and Young [6], Manning [7], Maunder [8], Neville [9], Rohrmann and Santos [10], Rucker [11], Maeda et al. [12], Sloane [13], Sommerville [14], Wells et al. [15] Nowadays, the research of hyperspherical functions is given both in Euclid's and
Riemann's geometry and topology (Riemann's and Poincare's sphere) multidimensional
potentials, theory of fluids, nuclear physics, hyperspherical black holes, and so
forth.

2. Hypersphere Function with Two Degrees of Freedom

The former results (see [4–30]) as it is known present two-dimensional (surface-surfs ), respectively, three-dimensional
(volume-solids) geometrical entities. In addition to certain generalizations [27], there exists a family of hyperspherical functions that can be presented in the
simplest way through the hyperspherical matrix , with two degrees of freedom and (), instead of the former presentation based only on vector approach (on the degree
of freedom k). This function is based on the general value of integrals, and so we obtain it's
generalized form.

Definition 2.1.

The generalized hyperspherical function is defined by

(21)

where is the gamma function.

On this function, we can also perform "motions'' to the lower degrees of freedom by
differentiating with respect to the radius ,starting from the th, on the basis of recurrence

(22)

The example of the spherical functions derivation is shown in Figure 1. The second () degree of freedom is achieved, and it is one level lower than the volume level (). Several fundamental characteristics are connected to the sphere. With its mathematically
geometrical description, the greatest number of information is necessary for a solid
sphere as a full spherical body. Then we have the surface sphere or surf-sphere, and
so forth.

Figure 1.Moving through the vector of real surfaces (left column), deducting one degree of
freedom of the surface sphere we obtain the circumference, and for two (degrees) we get the
point 2. Moving through the vector of real solids (right column), that is, by deduction of
one degree of freedom from the solid sphere, we obtain a circle (disc), and for two (degrees of freedom),
we obtain a line segment or diameter.

The appropriate matrix () is formed on the basis of the general hyperspherical function, and here it gives
the concrete values for the selected submatrix () as shown in Figure 2.

Figure 2.The submatrix of the function that covers one area of real degrees of freedom (). Also noticeable are the coordinates of six sphere functions (undef. are nondefined, predominantly of singular value, and 0 are zeros of this function).

3. Translators in the Matrix Conversion of Functions

A more generalized relation which would connect every element in the matrix, (Figures
3 and 4) both discrete and/or continual ones, can be defined on the basis of relation (quotient)
of two hyperspherical functions, one with increment of arguments dimensions—degrees
of freedom (), that is, the assigned one—while the other would be the starting one (the referred
one). On the basis of the previous definition, the translator is

(31)

Figure 3.The position of the reference element and its surrounding in the hypersphere matrix
when

Figure 4.The example of the position of the referent and assigned element (of the nested HS
function in the block-matrix) in the space of the selected hyperspheric block-matrix.

Note.

In the previous expression we do not take into consideration the radius increase as
a degree of freedom, so . The defining function thus equals

(32)

This equation can be expressed in the form

(33)

Every matrix element as a referring one can have in total eight elements in its neighbourhood,
and it makes nine types of connections (one with itself) in the matrix plane (Figure
3). Considering that two degrees of freedom have a positive or negative increment (in
this case integer), the selected submatrix is representative enough from the aspect
of the functions conversion in plane with the help of the translator .

Since the accepted increments are threefold , and connections are established only between the two elements, the number of total
connections in the representative submatrix is . All relations of this submatrix and the translators that enable those connections
can now be summarised in Table 1.

4. Generalized Translators of the Hyperspherical Matrix

In this section the extended recurrent operators include one more dimension as a degree
of freedom, which is the radius r. If the increment and/or reduction is applied on this argument as well, the translator
will get the extended form

Definition 4.1.

The translator is defined with

(41)

Since all the three increments can be positive or in symmetrical case negative (in
reduction), expression (3.1) is extended and is reduced to relation (4.1), where the
differences are the increment values or reduction of the variables and in relation to the referent coordinate (-function) in the matrix. All three variables can be real numbers (). The obtained expression is a more generalized translator, a more generalized functional operator, and in
that sense it will be defined as the generalized translator. When, besides the unitary
increments of the degree of freedom and , we introduce the radius increment the number of combinations becomes exponential (Table 2), and it is .

The schematic presentation of "3D motions'' through the space of block-submatrix and
locating the assigned HS function on the basis of translators and the starting hyperspherical
function is given in Figure 4.

According to the translator that includes three arguments, and in view of it "covering the field'' of the block-submatrix
according to Figure 4, we have the following function:

(42)

Examples 4.2.

Depending on which level we observe, the translator can be applied on block-matrix,
matrix, or vector relations. The most common is the recurrent operator for the elements
of columns' vector. Then, it is usually reduced onto one variable and that is the
degree of freedom . If the increment is an integer and , the recurrent relation for any element of the nth column is defined as

(43)

If the relation is restricted to , that is, on the vector particular to this degree of freedom and on the increment
, the translator can be simplified as

(44)

For the unit radius this expression can be reduced to the relation

(45)

If we analyze the relation for the vector with the degree of freedom and the increment , the translator now becomes

(46)

On the basis of the previous positions and results, we define two recurrent operators
for defining the assigned functions

As the recurrent operator is generalized, the increments can also be negative (ones)
and noninteger; so, for example, for the block matrix recursion, with the selected
increments , and , the operator has a more complex structure

(49)

5. Conversion of the Basic Spheric Entities

Example 5.1.

All relations among the six real geometrical sphere entities are presented on the
basis of the translator . These entities are P-point, D-diameter, C-circumference, A-circle, S-surface, and V-sphere volume, given in Figure 5. In addition to the graph presentation, the relation among these entities can also
be a graphical one, as shown in Figure 5.

6. The Relation of a Point and Real Spherical Entities

A point is a mathematical notion that from the epistemological standpoint has great
theoretical and practical meaning. Here, a point is a solid-sphere of which two degrees
of freedom of type and one of type are reduced. Of course, a point can be also defined in a different way, which
has not been analyzed in the previous procedures. Here these relations are considered
separately, and therefore we develop specific translators of the type. On the basis of the established graph, all option relations are shaped. There
are in total thirty of them (), and they are presented in Table 3.

The previous operators can form a relation among six real spherical entities. From
a formal standpoint some of them can be shaped as well on the basis of the beta function,
if we include the following relation originating from Legendre

(61)

The position of the six analyzed coordinates of the real spherical functions can be
presented on the surface hyperspherical function 3D (Figure 6, using the software Mathematica).

Figure 6.The positions of spherical entities on 3D graphic of the HS function of unit radius.

7. Conclusion

The hyperspherical translators have a specific role in establishing relations among
functions of some spherical entities. Meanwhile, their role is also enlarged, because
this relation can be analytically expanded to the complex part of the hyperspherical
function (matrix). In addition, there can be increments of degrees of freedom with
noninteger values. The previous function properties of translator functions are provided
thanks to the interpolating and other properties of the gamma function. Functional
operators defined in the previously described way can be applied in defining the total
dimensional potential of the hyperspherical function in the field of natural numbers
(degrees of freedom ). Namely, this potential (dimensional flux) can be defined with the double series:

(71)

Here, the translators are applied taking into consideration that every defining function
can be presented on the basis of the reference HS function, if we correctly define
the recurrent relations both for the series and for the columns of the hyperspherical
matrix [27].

References

Bishop, M, Whitlock, PA: The equation of state of hard hyperspheres in four and five dimensions. Journal of Chemical Physics. 123(1), (2005)